Extended Tensor Train (ETT)

• Extended Tensor Train (ETT) is a tensor-network representation that augments the classical tensor train by separating mode-wise linear factors from a low-rank TT core via a two-level Tucker-TT factorization.
• ETT employs discrete and functional extensions—including FTT, STT, and EFTT—to reduce storage costs and enhance approximation accuracy for large-scale tensor problems.
• Advanced variants such as shared-factor ETT use Riemannian optimization to manage multilinear manifolds, enabling improved performance in applications like image restoration and eigenvalue computations.

Searching arXiv for papers on “Extended Tensor Train” and related functional/manifold formulations. I’m checking arXiv for papers specifically using the term “Extended Tensor Train”, as well as related “extended functional tensor train” and “spectral tensor-train” work. Extended Tensor Train (ETT) denotes a class of tensor-network representations that enlarge the classical tensor train (TT) format by separating mode-wise linear factors from a low-rank TT-structured core. In the discrete setting, this is a two-level factorization: a tensor is first represented in Tucker form and the Tucker core is then represented in TT form. In the functional setting, closely related developments extend TT from discrete arrays to multivariate functions through functional tensor-train (FTT), spectral tensor-train (STT), and extended functional tensor train (EFTT) constructions. The term is therefore not uniform across the literature: some papers use “ETT” explicitly for the Tucker-plus-TT construction, whereas earlier work develops mathematically similar extensions of TT without introducing that acronym (Strössner et al., 2022, Molozhavenko et al., 28 Aug 2025, Bigoni et al., 2014).

1. Classical tensor train as the baseline model

The standard TT decomposition represents an order-$d$ tensor $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$ by a chain of 3-way cores $G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$ with $r_0=r_d=1$: $$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$ Its storage complexity is

$$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$

which becomes $\mathcal{O}(dnr^2)$ in the isotropic case $n_k=n$, $r_k=r$. TT-SVD provides a quasi-optimal low-rank approximation with a global error bound; if each unfolding is truncated at tolerance $\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$, then the resulting approximation satisfies

$A \in \mathbb{R}^{n_1\times\cdots\times n_d}$0

These properties explain why TT is the point of departure for essentially all later “extended” variants (Bigoni et al., 2014).

The 2016 algorithmic literature treats TT interchangeably with matrix product states (MPS) and emphasizes its orthogonality structure. If $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$1 is the matricization of a core over its first two modes and $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$2 over its last two modes, then left- and right-orthogonality are

$A \in \mathbb{R}^{n_1\times\cdots\times n_d}$3

This stabilizes contractions, localizes Frobenius norms, and underlies alternating local-update algorithms. A common misconception is that ETT replaces TT; more precisely, ETT builds on TT and inherits its rank, gauge, and contraction machinery (Phan et al., 2016).

2. ETT as a two-level Tucker-TT factorization

In the discrete tensor setting, ETT augments a standard TT representation by inserting low-dimensional factor matrices per mode, typically obtained from a Tucker decomposition, and then representing the Tucker core itself in TT format. Concretely, a large coefficient tensor is approximated as

$A \in \mathbb{R}^{n_1\times\cdots\times n_d}$4

where $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$5 is a small core of size $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$6 and each $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$7 is tall; the core $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$8 is then approximated in TT with TT cores of sizes $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$9. This two-level factorization reduces storage and construction cost when $G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$0 (Strössner et al., 2022).

The 2025 formulation makes this structure explicit. For $G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$1, let the Tucker ranks be $G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$2 and the TT ranks be $G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$3 with $G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$4. The Tucker core $G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$5 has TT cores

$G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$6

and the ETT decomposition is

$G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$7

where $G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$8 denotes the TT contraction of the core and $G_k \in \mathbb{R}^{r_{k-1}\times n_k\times r_k}$9 are Tucker factors. In this sense, ETT separates the physical indices via Tucker factors while representing the Tucker core in TT form. The resulting format is a special case of Hierarchical Tucker (HT), but with the linear-chain structure and algorithmic simplicity of TT (Molozhavenko et al., 28 Aug 2025).

This two-level architecture clarifies what is meant by “extended” in ETT. The extension is structural rather than merely algorithmic: large physical mode sizes are absorbed into factor matrices, while the residual inter-mode coupling is compressed by a TT core. A plausible implication is that ETT is most natural when the original mode sizes are large but the multilinear ranks are moderate.

3. Shared-factor ETT and manifold structure

A more specialized variant is ETT with shared factors, denoted SF-ETT. Here the modes are partitioned into $r_0=r_d=1$0 non-shared modes and $r_0=r_d=1$1 shared modes. For the non-shared modes $r_0=r_d=1$2, one uses $r_0=r_d=1$3; for the shared modes $r_0=r_d=1$4, the same factor

$r_0=r_d=1$5

is repeated. The decomposition becomes

$r_0=r_d=1$6

The associated SF-ETT ranks are

$r_0=r_d=1$7

$r_0=r_d=1$8

$r_0=r_d=1$9

These definitions couple TT-style unfoldings and Tucker-style matricizations in a single rank specification (Molozhavenko et al., 28 Aug 2025).

The shared-factor constraint changes the optimization landscape. In ordinary Tucker or ETT form, the mapping from factors to the represented tensor is multilinear in each block separately. When the same $$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$0 appears in several modes, that multilinearity is broken, because $$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$1 occurs repeatedly in the contraction. Consequently, alternating least squares no longer produces a sequence of independent linear subproblems. The 2025 work therefore adopts a Riemannian viewpoint and proves that the SF-ETT set is a smooth embedded submanifold, obtained as the transversal intersection

$$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$2

inside the Tucker manifold $$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$3. Its dimension is

$$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$4

with

$$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$5

This gives ETT a geometric status absent from ad hoc rank-constrained parameterizations (Molozhavenko et al., 28 Aug 2025).

4. Functional extensions: FTT, STT, and EFTT

The earliest continuous extension relevant to ETT is the functional tensor-train decomposition. Let $$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$6 with product measure $$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$7, and let $$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$8. The FTT representation writes

$$A_{i_1,\dots,i_d} = \sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}} G_1(1,i_1,\alpha_1)\,G_2(\alpha_1,i_2,\alpha_2)\cdots G_d(\alpha_{d-1},i_d,1).$$9

where $$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$0 and the entries of each $$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$1 are univariate functions. In index form,

$$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$2

This construction is obtained recursively from Schmidt decompositions of Hilbert–Schmidt kernels. If the FTT is truncated at ranks $$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$3, the residual satisfies

$$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$4

Under Sobolev regularity and Hölder continuity assumptions, the decomposition converges absolutely almost everywhere, and the cores inherit Sobolev regularity (Bigoni et al., 2014).

The spectral tensor-train approximation then represents each functional core in a univariate orthogonal polynomial basis: $$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$5 or, equivalently,

$$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$6

The global STT approximation becomes

$$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$7

with storage

$$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$8

For $$\mathcal{O}\!\left(\sum_{k=1}^d n_k\,r_{k-1}r_k\right),$$9, pseudospectral projection yields polynomial convergence, and for analytic $\mathcal{O}(dnr^2)$0 one obtains exponential convergence in $\mathcal{O}(dnr^2)$1 (Bigoni et al., 2014).

EFTT, introduced in 2022, is the functional analogue of discrete ETT. Instead of parameterizing every univariate function $\mathcal{O}(dnr^2)$2 separately as in prior FTT approaches, it builds a shared basis $\mathcal{O}(dnr^2)$3 in each mode, forms a small core $\mathcal{O}(dnr^2)$4, and approximates that core in TT. For $\mathcal{O}(dnr^2)$5,

$\mathcal{O}(dnr^2)$6

with

$\mathcal{O}(dnr^2)$7

The corresponding evaluation tensor $\mathcal{O}(dnr^2)$8 is defined on tensorized Chebyshev grids, and the coefficient tensor is obtained by modewise discrete cosine transforms,

$\mathcal{O}(dnr^2)$9

The paper’s stability bound splits the total error into interpolation error and low-rank tensor approximation error: $n_k=n$0 A central point of terminology follows from this literature: the 2014 paper does not use the acronym “ETT,” but its FTT and STT constructions are natural instances of TT extended to function spaces; the 2022 paper then makes the “extended” interpretation explicit in the functional setting (Strössner et al., 2022, Bigoni et al., 2014).

5. Construction algorithms and optimization methods

Several distinct algorithmic traditions underlie ETT and related formats. For STT, the construction is non-intrusive and sampling-based. One chooses quadrature grids $n_k=n$1, forms a weighted sample tensor

$n_k=n$2

applies TT-DMRG-cross to obtain $n_k=n$3 with

$n_k=n$4

undoes the weights, and projects each TT core onto the chosen univariate basis. The method is rank-revealing, concentrates samples where needed, and empirically exhibits near-linear sampling complexity in $n_k=n$5 for analytically low-rank functions (Bigoni et al., 2014).

EFTT uses a different two-stage compression strategy. In phase 1, randomized ACA samples mode fibers of the evaluation tensor, QR is applied to the sampled fiber matrix, and DEIM selects interpolation indices. Polynomial degrees $n_k=n$6 are adjusted by a Chebfun chopping heuristic. In phase 2, the resulting small core $n_k=n$7 is approximated by the rank-adaptive greedy restricted cross algorithm, requiring only $n_k=n$8 evaluations of entries of $n_k=n$9. In storage terms, EFTT separates TT-core storage $r_k=r$0 from univariate basis storage $r_k=r$1, giving overall $r_k=r$2 storage (Strössner et al., 2022).

A separate algorithmic line concerns “extended TT” in the sense of local update rules rather than a new tensor format. The 2016 paper develops Alternating Multi-Cores Update (AMCU) methods that update one, two, or three consecutive TT cores per sweep. Its ASCU two-sides variant writes a core as

$r_k=r$3

solves a Tucker-2 subproblem locally, and absorbs $r_k=r$4 and $r_k=r$5 into neighboring cores. ADCU and ATCU perform overlapping two-core and three-core updates, respectively, in a DMRG-style scheme. This does not define ETT as a separate representation, but it does exemplify an extended-core viewpoint frequently associated with the term in later discussion (Phan et al., 2016).

For SF-ETT, the main computational framework is Riemannian optimization. Tangent vectors are parameterized by TT-core variations $r_k=r$6, non-shared factor variations $r_k=r$7, and a shared-factor variation $r_k=r$8, subject to gauge conditions such as

$r_k=r$9

Orthogonal projection of an ambient tensor $\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$0 onto the tangent space has closed-form expressions for $\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$1, $\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$2, and $\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$3; the retraction is defined by quasi-optimal SF-ETT-SVD rounding,

$\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$4

The quasi-optimality constant is

$\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$5

Automatic differentiation is used to compute tangent-space projections and vector transport efficiently (Molozhavenko et al., 28 Aug 2025).

6. Applications, empirical behavior, and limitations

The empirical record of extended TT methods is heterogeneous because the term covers several constructions. For STT, the reported numerical experiments include modified Genz functions with dimensions up to $\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$6, mixed Fourier-mode functions, local Gaussian bumps, and an elliptic PDE with random inputs. For smooth Genz functions, STT shows spectral convergence with increasing polynomial degree; for nonsmooth cases, rates drop to quadratic or slower. In analytically low-rank cases, TT ranks remain constant with dimension and the number of function evaluations scales roughly linearly in $\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$7. Against anisotropic adaptive Smolyak sparse grids with Gauss–Patterson rules, STT achieves much faster error decay, especially as $\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$8 increases (Bigoni et al., 2014).

EFTT reports a different type of gain: reduced sampling and storage relative to continuous FTT constructions. Compared to the algorithm of Gorodetsky, Karaman, and Marzouk, it reduces the number of function evaluations required to achieve a prescribed accuracy by up to over $\delta=\varepsilon/\sqrt{d-1}\,\|A\|_F$9, and on the Ackley function it typically reduces storage by over $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$00. Over a broader benchmark set, average reductions of around $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$01–$A \in \mathbb{R}^{n_1\times\cdots\times n_d}$02 in evaluations and storage relative to direct TT-cross on the evaluation tensor are reported, with larger savings when univariate fibers are strongly compressible (Strössner et al., 2022).

The AMCU line of work evaluates extended-core TT updates on denoising, blind source separation, and image restoration. In denoising exponentially decaying signals at $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$03 and SNR $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$04 dB, TT-SVD yields SAE approximately $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$05–$A \in \mathbb{R}^{n_1\times\cdots\times n_d}$06 dB, whereas AMCU variants yield SAE approximately $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$07–$A \in \mathbb{R}^{n_1\times\cdots\times n_d}$08 dB. For single-mixture blind source separation at SNR $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$09 dB, ASCU attains SAE approximately $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$10–$A \in \mathbb{R}^{n_1\times\cdots\times n_d}$11 dB per source with monotone decrease in global error. In image denoising, TT-ASCU outperforms TT-SVD and several competing tensor methods on MSE, PSNR, and SSIM; for Lena at SNR $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$12 dB, TT-ASCU reports MSE $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$13 dB, PSNR $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$14 dB, SSIM $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$15, versus TT-SVD MSE $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$16 dB and PSNR $A \in \mathbb{R}^{n_1\times\cdots\times n_d}$17 dB (Phan et al., 2016).

For SF-ETT, the reported applications are tensor approximation and multidimensional eigenvalue problems. In least-squares approximation, SF-ETT rounding already produces solutions close to local optima, and subsequent Riemannian steepest-gradient refinement gives only marginal improvements. In eigenvalue computations for the Laplace operator and the Henon–Heiles potential, average iteration time decreases as the number of shared factors increases, with especially pronounced gains in the more complex Henon–Heiles case (Molozhavenko et al., 28 Aug 2025).

The principal limitations are also consistent across the literature. TT ranks can grow with dimension, with local or nonsmooth features, or with poor variable ordering. Basis choice must match the measure and regularity, for example Legendre or Chebyshev on bounded intervals and Hermite for Gaussian measures. In EFTT, cumulative projection constants can erode accuracy in very high dimensions. In SF-ETT, imposing shared factors on dissimilar modes can degrade approximation quality or force larger ranks. A recurring misconception is that “ETT” names a single universally fixed format; the literature instead contains a family of related extensions, united by the idea of enriching TT with additional structural layers—functional cores, spectral bases, mode-wise factor matrices, shared factors, or multi-core local updates—while preserving TT’s low-rank chain structure (Bigoni et al., 2014, Strössner et al., 2022, Molozhavenko et al., 28 Aug 2025, Phan et al., 2016).